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Transcript of Ch01_ Introduction 2015
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Mechanical
VibrationsSingiresu S.
Rao
SI Edition
Chapter 1
Fundamentals of
Vibration
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© 2005 Pearson Education South Asia Pte Ltd. 2
1. Fundamentals of Vibration2. Free Vibration of Single !F S"stems
#. $armonicall" E%cited Vibration
&. Vibration under 'eneral Forcing(onditions
5. )*o !F S"stems
+. ,ultidegree of Freedom S"stems-. etermination of atural Fre/uencies
and ,ode Shaes
Course Outline
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© 2005 Pearson Education South Asia Pte Ltd. #
. (ontinuous S"stems
. Vibration (ontrol
10. Vibration ,easurement and Alications
Course Outline
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© 2005 Pearson Education South Asia Pte Ltd. &
1.1 Preliminar" 3emar4s
1.2 rief $istor" of Vibration
1.# 6mortance of the Stud" of Vibration
1.& asic (oncets of Vibration
1.5 (lassification of Vibration
1.+ Vibration Anal"sis Procedure1.- Sring Elements
(hater !utline
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© 2005 Pearson Education South Asia Pte Ltd. 5
1. ,ass or 6nertia Elements
1. aming Elements
1.10 $armonic ,otion
1.11 $armonic Anal"sis
(hater !utline
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© 2005 Pearson Education South Asia Pte Ltd. +
1.1 Preliminar" 3emar4s
7E%amination of 8ibration9s imortant role7 Vibration anal"sis of an engineering s"stem
7 efinitions and concets of 8ibration7 (oncet of harmonic anal"sis for general
eriodic motions
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© 2005 Pearson Education South Asia Pte Ltd. -
1.# 6mortance of the Stud" of Vibration
7:h" stud" 8ibration;Vibrations can lead to e%cessi8e deflections
and failure on the machines and structures)o reduce 8ibration through roer design of
machines and their mountings)o utili *elding rocesses)o stimulate earth/ua4es for geological
research and conduct studies in design ofnuclear reactors
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© 2005 Pearson Education South Asia Pte Ltd.
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© 2005 Pearson Education South Asia Pte Ltd.
1.& asic (oncets of Vibration
Vibration ? an" motion that reeats itself afteran inter8al of time
Vibrator" S"stem consists of@1 sring or elasticit"
2 mass or inertia# damer
6n8ol8es transfer of otential energ" to 4inetic
energ" and 8ice 8ersa
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© 2005 Pearson Education South Asia Pte Ltd. 10
1.& asic (oncets of Vibration
egree of Freedom (d.o.f.) ?min. no. of indeendent coordinates re/uiredto determine comletel" the ositions of allarts of a s"stem at an" instant of time
E%amles of single degreeBofBfreedoms"stems@
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© 2005 Pearson Education South Asia Pte Ltd. 11
1.& asic (oncets of Vibration
E%amles of single degreeBofBfreedoms"stems@
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© 2005 Pearson Education South Asia Pte Ltd. 12
1.& asic (oncets of Vibration
E%amles of )*o degreeBofBfreedom s"stems@
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© 2005 Pearson Education South Asia Pte Ltd. 1#
1.& asic (oncets of Vibration
E%amles of )hree degree of freedom s"stems@
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© 2005 Pearson Education South Asia Pte Ltd. 15
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© 2005 Pearson Education South Asia Pte Ltd. 1+
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© 2005 Pearson Education South Asia Pte Ltd. 1-
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© 2005 Pearson Education South Asia Pte Ltd. 1
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© 2005 Pearson Education South Asia Pte Ltd. 1
1.5 (lassification of Vibration
Free Vibration@ A s"stem is left to 8ibrate on its o*n after aninitial disturbance and no e%ternal force acts onthe s"stem. E.g. simle endulum
Forced Vibration@ A s"stem that is subCected to a reeatinge%ternal force. E.g. oscillation arises from dieselengines
Resonance occurs *hen the fre/uenc" of thee%ternal force coincides *ith one of thenatural fre/uencies of the s"stem
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© 2005 Pearson Education South Asia Pte Ltd. 20
1.5 (lassification of Vibration
Dndamed Vibration@:hen no energ" is lost or dissiated in frictionor other resistance during oscillations
amed Vibration@
:hen any energ" is lost or dissiated infriction or other resistance during oscillations
Linear Vibration@
:hen all basic comonents of a 8ibrator"s"stem= i.e. the sring= the mass and thedamer beha8e linearl"
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© 2005 Pearson Education South Asia Pte Ltd. 21
1.+ Vibration Anal"sis Procedure
Ste 1@ ,athematical ,odeling
Ste 2@ eri8ation of 'o8erning E/uations
Ste #@ Solution of the 'o8erning E/uations
Ste &@ 6nterretation of the 3esults
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© 2005 Pearson Education South Asia Pte Ltd. 22
1.+ Vibration Anal"sis Procedure
E%amle of the modeling of a forging hammer@
E l 1 1
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© 2005 Pearson Education South Asia Pte Ltd. 2#
E%amle 1.1,athematical ,odel of a ,otorc"cle
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© 2005 Pearson Education South Asia Pte Ltd. 2&
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© 2005 Pearson Education South Asia Pte Ltd. 25
Dsing mathematical model to reresent theactual 8ibrating s"stemE.g. 6n figure belo*= the mass and daming
of the beam can be disregarded the s"stem
can thus be modeled as a sringBmasss"stem as sho*n.
1. ,ass or 6nertia Elements
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© 2005 Pearson Education South Asia Pte Ltd. 2+
1. aming Elements
Viscous aming@
aming force is roortional to the 8elocit" ofthe 8ibrating bod" in a fluid medium such as air=*ater= gas= and oil.
(oulomb or r" Friction aming@aming force is constant in magnitude butoosite in direction to that of the motion of the8ibrating bod" bet*een dr" surfaces
,aterial or Solid or $"steretic aming@Energ" is absorbed or dissiated b" materialduring deformation due to friction bet*eeninternal lanes
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© 2005 Pearson Education South Asia Pte Ltd. 2-
E%amle 1.10 E/ui8alent Sring and aming(onstants of a ,achine )ool Suort
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© 2005 Pearson Education South Asia Pte Ltd. 2
E%amle 1.10 Solution
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© 2005 Pearson Education South Asia Pte Ltd. 2
Let the total forces acting on all the srings and allthe damers be F s and F d = resecti8el" see Fig.1.#-d. )he force e/uilibrium e/uations can thusbe e%ressed as
E%amle 1.10 Solution
E.1)(4,3,2,1;
4,3,2,1;
==
==
i xc F
i xk F
idi
i si
E.2)(4321
4321
d d d d d
s s s s s
F F F F F
F F F F F
+++=
+++=
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© 2005 Pearson Education South Asia Pte Ltd. #0
E%amle 1.10 Solution
E.3)( xc F
xk F
eqd
eq s
=
=
*here F s
+ F d
= W = *ith W denoting the total8ertical force including the inertia force acting onthe milling machine. From Fig. 1.#-d= *e ha8e
E/uation E.2 along *ith E/s. E.1 and E.#=
"ield
E.4)(4
4
4321
4321
cccccc
k k k k k k
eq
eq
=+++=
=+++=
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© 2005 Pearson Education South Asia Pte Ltd. #1
( )1.1kx F =
F ? sring force=k ? sring stiffness or sring constant= and x ? deformation dislacement of one end
*ith resect to the other
1.- Sring Elements
Linear sring is a t"e of mechanical lin4 that isgenerall" assumed to ha8e negligible mass anddaming
Spring force is gi8en b"@
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© 2005 Pearson Education South Asia Pte Ltd. #2
Work done (U) in deforming a sring or thestrain otential energ" is gi8en b"@
( )2.12
1 2kxU =
1.- Sring Elements
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© 2005 Pearson Education South Asia Pte Ltd. ##
1.- Sring Elements
Static deflection of a beam at the free end is
gi8en b"@
Spring onstant is gi8en b"@
( )6.13
3
EI
Wl st =δ
W ? mg is the *eight of the mass m= E ? Goung9s ,odulus= and I ? moment of inertia of crossBsection of beam
( )7.13
3
l
EI W k
st
==
δ
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© 2005 Pearson Education South Asia Pte Ltd. #&
1.- Sring Elements
(ombination of Srings@
!) Springs in parallel H if *e ha8e n sringconstants k 1= k 2= I= k n in parallel = then thee/ui8alent sring constant k e/ is@
( )11.121 ... neq k k k k +++=
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© 2005 Pearson Education South Asia Pte Ltd. #5
1.- Sring Elements
(ombination of Srings@
") Springs in series H if *eha8e n sring constants k 1=k 2= I= k n in series= then the
e/ui8alent sring constant4e/ is@
( )17.11
...
111
21 neqk k k k +++=
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© 2005 Pearson Education South Asia Pte Ltd. #+
( )30.1sinsin t A A x ω θ ==
( )31.1cos t Adt
dxω ω =
( )32.1sin 222
2
xt Adt
xd ω ω ω −=−=
1.10 $armonic ,otion
Periodic ,otion@ motion reeated after e/ual
inter8als of time$armonic ,otion@ simlest t"e of eriodic
motion
islacement x@ (on #ori$ontal a%is)
Velocit"@
Acceleration@
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© 2005 Pearson Education South Asia Pte Ltd. #-
(omle% number reresentation of harmonic
motion@
*here i ? J –1) and a and b denote the real and
imaginar" x and y comonents of & =resecti8el".
( )35.1iba X +=
1.10 $armonic ,otion
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© 2005 Pearson Education South Asia Pte Ltd. #
efinitions of )erminolog"@
Amlitude A is the ma%imum dislacementof a 8ibrating bod" from its e/uilibriumosition
Period of oscillation T is time ta4en tocomlete one c"cle of motion
Fre/uenc" of oscillation f is the no. ofc"cles er unit time
( )59.12
ω
∏
=T
( )60.12
1
π
ω ==
T f
1.10 $armonic ,otion
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1.10 $armonic ,otion
efinitions of )erminolog"@
atural fre/uenc" is the fre/uenc" *hich as"stem oscillates *ithout e%ternal forces
Phase angle φ is the angular differencebet*een t*o s"nchronous harmonic motions
( )
( ) ( )62.1sin
61.1sin
22
11
φ ω
ω
+=
=
t A x
t A x